This study estimated the maximum allowable system latency for haptic displays that produce tactile stimuli in response to the hand movements of users. In Experiment 1, two types of detection thresholds were estimated for the time delay of stimuli through psychophysical experiments involving 13 participants. One was a threshold for the users to notice the existence of a time delay. The other was a threshold for the users to experience changes in the perceived textures in comparison with stimuli with no time delay. The estimated thresholds were approximately 60 and 40 ms, respectively. In interviews, the participants reported that they experienced various types of subjective changes due to the time delay. In Experiment 2, the types of subjective sensations that might be altered by the time delay were investigated. The time delays were controlled based on the acceleration of the hand motions of the participants. The participants evaluated the differences in the perceived textures between the stimuli with a controlled time delay and ones with no delay. The results indicated that the participants associated the time-delayed stimuli with changes in mechanical parameters such as kinetic friction coefficient in addition to changes in the perceived roughness of the textures.